Theorem reseq1d 4946

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
reseq1d	|- ( ph -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq1 4941	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1364   |` cres 4666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-res 4676

This theorem is referenced by:  reseq12d  4948  fun2ssres  5302  funcnvres2  5334  funimaexg  5343  fresin  5439  offres  6201  tfrlemisucaccv  6392  tfrlemi1  6399  tfr1onlemsucaccv  6408  tfrcllemsucaccv  6421  freceq1  6459  freceq2  6460  fseq1p1m1  10186  setsresg  12741  setscom  12743  znle2  14284  dvcoapbr  15027  bj-charfundcALT  15539